But it is not necessary that the differentiation with respect to real and imaginary axis had to be same, right? I mean it's like partial derivative where derivative with respect to one axes (real in this case) need not be same as the derivative with respect to another axes (imaginary axes).

But it is not necessary that the differentiation with respect to real and imaginary axis had to be same, right? I mean it's like partial derivative where derivative with respect to one axes (real in this case) need not be same as the derivative with respect to another axes (imaginary axes).

thanks a lot for your help.

But you are finding a limit; the derivative is a limit , the quotient limit

[f(z+zo)-f(z)]/(z-zo) as z→zo . But z can approach zo along _every possible

complex direction. Then f'(z) exists when this limit exist,so that the limit must

exist along any direction along which you approximate zo, and for the limit to

exist, it must be the same no-matter how you approximate zo. In particular,

(re Cauchy-Riemann) , the approximation along the x-axis and the y-axis must